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\title{Preconditioning for eigenproblems}
\author{Gerard Sleijpen \\
       Utrecht University}
\date{}
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\subsection*{Abstract}
In the Jacobi-Davidson methods for eigenproblems a correction
equation for the expansion of the search subspace is an important
key for success. If the correction equation is solved accurately
then we will have quadratic convergence of the eigenvalues, but
unfortunately accurate solution by direct techniques will be far too
expensive in actual situations.

The usual approach is to solve the
correction equation only approximately by some suitable
method, in particular by a few steps of a preconditioned iterative
solution method. Because of the projections that occur in the correction
equation the inclusion of preconditioning is not so straightforward
and we will discuss ways of including preconditioning in an efficient
and stable manner. For accurate preconditioners as MRILU an equivalent
formulation of the correction equation as an augmented matrix without
projections may be more attractive.

In order to get flawless convergence one has to make the right selections
for the approximate eigenvectors. This and other
aspects that lead to an enhance performance of
Jacobi-Davidson will be discussed as well.

\medskip
This is joint work with Henk van der Vorst (Utrecht University)
and Freddy Wubs (University of Groningen).

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Gerard Sleijpen\\
Department of Mathematics\\
Utrecht University\\
PO Box 80 010\\
NL-3508 TA  Utrecht\\
 The Netherlands\\
{\tt http://www.math.ruu.nl/people/sleijpen/}
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